If you don't really want a filter, a simple loop would be easier I think
	Option Explicit
Sub Clearme()
'
' Clearme Macro
' Clear all filtered data
'
' Keyboard Shortcut: Ctrl+Shift+C
'
    Worksheets("Interface").Range("C9:I40").ClearContents
    
End Sub
Sub Filterme()
    
    Dim wsData As Worksheet, wsInterface As Worksheet
    
    Dim dtStart As Date, dtEnd As Date
    Dim sCustomer As String, sProduct As String
    Dim iInvoice As Long
    Dim iOut As Long, iDataRow As Long
    
    Set wsData = Worksheets("Data")
    Set wsInterface = Worksheets("Interface")
    
    Application.ScreenUpdating = False
    
    With wsInterface
        .Range("C9:I40").ClearContents
        dtStart = .Range("C6").Value
        dtEnd = .Range("D6").Value
        sCustomer = LCase(.Range("E6").Value)
        iInvoice = .Range("F6").Value
        sProduct = LCase(.Range("G6").Value)
    End With
    If CLng(dtStart) = 0 Then dtStart = DateSerial(2000, 1, 1)
    If CLng(dtEnd) = 0 Then dtEnd = DateSerial(2099, 12, 31)
    iOut = 9
    iDataRow = 5
    
    With wsData
        'Date    Invoice Number  Customer    Unit price  Product Name    Quantity    Total
        Do While Len(.Cells(iDataRow, 4).Value) > 0
            If .Cells(iDataRow, 4).Value < dtStart Then GoTo NextDataRow
            If .Cells(iDataRow, 4).Value > dtEnd Then GoTo NextDataRow
            If iInvoice > 0 Then
                If .Cells(iDataRow, 5).Value <> iInvoice Then GoTo NextDataRow
            End If
            If Len(sCustomer) > 0 Then
                If LCase(.Cells(iDataRow, 6).Value) <> sCustomer Then GoTo NextDataRow
            End If
            If Len(sProduct) > 0 Then
                If LCase(.Cells(iDataRow, 8).Value) <> sProduct Then GoTo NextDataRow
            End If
                        
            Call .Cells(iDataRow, 4).Resize(1, 7).Copy(wsInterface.Cells(iOut, 3))
                        
            iOut = iOut + 1
            
NextDataRow:
            iDataRow = iDataRow + 1
        
        Loop
    End With
     
    Application.ScreenUpdating = True
     
    MsgBox "Done"
     
     
End Sub
 
Attached is my update if you're interested
You should add error checking, e.g. is Start really a date, etc.